If I dissolve a Tums tablet #("1000 mg CaCO"_3)# in #"25 mL"# of water, how do I calculate the #"pH"# of the resultant suspension? When done experimentally, it was between #9# and #10#
1 Answer
Here's what I got.
Explanation:
There are several important things to keep in mind here. For starters, calcium carbonate is considered insoluble in water.
A saturated solution of calcium carbonate will hold about
So assuming that your solution is at
#25 color(red)(cancel(color(black)("mL water"))) * "0.013 g CaCO"_3/(10^3color(red)(cancel(color(black)("mL water")))) = "0.000325 g CaCO"_3#
of dissociated calcium carbonate. Use the molar mass of calcium carbonate to convert this to moles
#0.000325 color(red)(cancel(color(black)("g"))) * "1 mole CaCO"_3/(100.09 color(red)(cancel(color(black)("g")))) = 3.25 * 10^(-6) quad "moles CaCO"_3#
Now, the calcium carbonate that will dissociate will do so to produce calcium cations and carbonate anions in
#"CaCO"_ (3(s)) rightleftharpoons "Ca"_ ((aq))^(2+) + "CO"_ (3(aq))^(2-)#
This means that the saturated solution of calcium carbonate will contain
The molarity of the carbonate anions will be equal to
#["CO"_3^(2-)] = (3.25 * 10^(-6)quad "moles")/(25 * 10^(-3) quad "L") = 1.30 * 10^(-4) quad "M"#
A second important thing to keep in mind here is that the carbonate anions will act as a weak base and react with water to form bicarbonate anions and hydroxide anions
#"CO"_ (3(aq))^(2-) + "H"_ 2"O"_ ((l)) rightleftharpoons "HCO"_ (3(aq))^(-) + "OH"_ ((aq))^(-)#
The acid dissociation constant for the bicarbonate anions is equal to--see here.
#K_ ("a HCO"_ 3^(-)) = 4.8 * 10^(-11)#
which means that the base dissociation constant of the carbonate anions,
#K_ ("b CO"_ 3^(2-)) = (1 * 10^(-14))/(4.8 * 10^(-11)) = 2.08 * 10^(-4)# This is the case because an aqueous solution at
#25^@"C"# has#K_a * k_b = 1 * 10^(-14)# .
By definition, the base dissociation constant is equal to
#K_ ("b CO"_ 3^(2-)) = (["HCO"_3^(-)] * ["OH"^(-)])/(["CO"_3^(2-)])#
If you take
#["HCO"_3^(-)] = ["OH"^(-)] = x quad "M"#
and
#["CO"_3^(2-)] = (1.30 * 10^(-4) - x) quad "M"# This is the case because in order for the reaction to produce
#x# #"M"# of hydroxide anions and of bicarbonate anions, it must consume#x# #"M"# of carbonate anions.
Plug this into the expression you have for the base dissociation constant to get
#K_ ("b CO"_ 3^(2-)) = (x * x)/(1.30 * 10^(-4) - x)#
#2.08 * 10^(-4) = x^2/(1.30 * 10^(-4) -x)#
Rearrange to quadratic equation form
#x^2 + 2.08 * 10^(-4) * x - 2.08 * 1.30 * 10^(-8) = 0#
This quadratic equation will produce two solutions, one positive and one negative, but since
#x = 9.06 * 10^(-5)#
This means that, at equilibrium, the resulting solution has
#["OH"^(-)] = 9.06 * 10^(-5) quad "M"#
Finally, the
#"pH" = 14 - [-log(["OH"^(-)])]#
#"pH"= 14 + log(9.06 * 10^(-5))#
#color(darkgreen)(ul(color(black)("pH" = 9.96)))#
I'll leave the answer rounded to two decimal places, the number of sig figs you have for the volume of water.